Chapter 4 Linear Equations and Inequalities in Two Variables

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Chapter 4Linear Equations and Inequalities in
Two Variables
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4.1
Graphing Linear Equations in Two Variables
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The Rectangular Coordinate System
In the rectangular coordinate system, the
horizontal number line is the x-axis. The
vertical number line is the y-axis. The point of
intersection of these axes is their zero points,
called the origin. The axes divide the plane into
4 quarters, called quadrants.
y-axis
1st quadrant
2nd quadrant
x-axis
3rd quadrant
4th quadrant
Blitzer, Introductory Algebra, 5e Slide 3
Section 4.1
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The Rectangular Coordinate System
Each point in the rectangular coordinate system
corresponds to an ordered pair of real numbers
(x,y). Note the word ordered because order
matters. The first number in each pair, called
the x-coordinate, denotes the distance and
direction from the origin along the x-axis. The
second number, the y-coordinate, denotes vertical
distance and direction along a line parallel to
the y-axis or along the y-axis Itself.
Point Movement from origin
(3.5) 3 units right and 5 units up
(4,-3) 4 units right and 3 units down
(-2,-7) 2 units left and 7 units down
(0,0) 0 right or left and 0 up or down
In plotting points, we move across first (either
left or right), and then move either up or down,
always starting from the origin.
Blitzer, Introductory Algebra, 5e Slide 4
Section 4.1
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Plotting Points
EXAMPLE
Plot the points (3,2) and (-2,-4).
SOLUTION
(3,2)
(-2,-4)
Blitzer, Introductory Algebra, 5e Slide 5
Section 4.1
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The Graph of an Equation
The graph of an equation in two variables is the
set of points whose coordinates satisfy the
equation. An ordered pair of real numbers (x,y)
is said to satisfy the equation when
substitution of the x and y coordinates into the
equation makes it a true statement. For
example, in the equation y 2x 6, the ordered
pair (1,8) is a solution. When we substitute this
point the sentence reads 8 8, which is
true. The ordered pair (2,3) is not a solution.
When we substitute this point, the sentence reads
3 10, which is not true.
Blitzer, Introductory Algebra, 5e Slide 6
Section 4.1
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Remember.
Solution of an Equation in Two Variables

• If when the x-coordinate of the ordered pair is
  substituted for x and the y-coordinate of the
  ordered pair is substituted for y, we obtain a
  true statement- then the ordered pair is a
  solution to the equation.

Blitzer, Introductory Algebra, 5e Slide 7
Section 4.1
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Solution of an Equation in Two Variables

• Example
• Given the equation 2x 3y 18, determine if
  the ordered pair (3, 4) is a solution to the
  equation.
• We substitute 3 in for x and 4 in for y.
• 2(3) 3 (4) ? 18
• 6 12 ? 18
• 18 18 True.
• Therefore, the ordered pair (3, 4) is a solution
  to the equation 2x 3y 18.

Blitzer, Introductory Algebra, 5e Slide 8
Section 4.1
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Finding Solutions of an Equation

• Find five solutions to the equation y 3x 1.
• Start by choosing some x values and then
  computing the corresponding y values.
• If x -2, y 3(-2) 1 -5. Ordered pair
  (-2, -5)
• If x -1, y 3(-1) 1 -2. Ordered pair
  ( -1, -2)
• If x 0, y 3(0) 1 1. Ordered pair
  (0, 1)
• If x 1, y 3(1) 1 4. Ordered pair
  (1, 4)
• If x 2, y 3(2) 1 7. Ordered pair
  (2, 7)

Blitzer, Introductory Algebra, 5e Slide 9
Section 4.1
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Graph of the Equation

• Plot the five ordered pairs to obtain the graph
  of y 3x 1

(2,7)
(1,4)
(0,1)
(-1,-2)
(-2,-5)
Blitzer, Introductory Algebra, 5e Slide 10
Section 4.1
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Graphing an Equation
EXAMPLE
Graphs in the rectangular coordinate system can
also be used to tell a story. Try to select the
graph that best illustrates the story of the
population of the U.S.A.
(a)
(c)
(b)
Population
Population
Population
Years
Years
Years
SOLUTION
Graph (c)
Blitzer, Introductory Algebra, 5e Slide 11
Section 4.1
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Linear Equations

• Definition of a Linear Equation
• A linear equation in one variable x is an
  equation that can be written in the form
  ax b 0, where a and b are real numbers and a
  is not equal to 0.
• An example of a linear equation in x is 4x 2
  6. Linear equations in x are first degree
  equations in the variable x.

Blitzer, Introductory Algebra, 5e Slide 12
Section 4.1
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Comparison of Graphs of Linear Equations

• The graph of y mx b is the graph of
• y mx shifted b units up when b is a positive
  number.
• The graph of y mx b is the graph of
• y mx shifted b units down when b is a negative
  number.
• More on lines to come in this chapter.

Blitzer, Introductory Algebra, 5e Slide 13
Section 4.1